The Short Version
The claim is largely accurate. Multiple veterinary dermatologists and biomedical sources confirm that animals — particularly dogs and cats — can develop allergic reactions to human dander (shed skin cells and hair proteins). The underlying immune mechanisms are well-established. However, such allergies appear to be uncommon, prevalence figures vary widely depending on the study population, and diagnostic testing has limitations. The claim is valid but would benefit from noting that these reactions are rare and specific to human dander rather than to humans broadly.